There is a clear positive correlation between disc size and luminosity with BH mass. We fit the data in the upper two panels of Fig. REF  with a straight line taking the form: \(Y=a+b[X-8.7]\) , where \(X\) =log(\(M_{\rm BH}\) ), and \(Y\) =log(\(R_{\rm 1/2}\) ). We normalized \(M_{\rm BH}\)  at \(=5\times 10^8\,\rm M_{\odot }\)  (i.e. log(\(M_{\rm BH}\) )=8.7) to minimize the error on \(a\)  and \(b\) . The fit was done in the log–log space, using the fiteyx routine of [1]}, which takes into account the error on both variables. The best-fit results in case (a) are: \(a_{\rm 1/2,obs}\) =15.89\(\pm 0.11\) , \(b_{\rm 1/2,obs}\) =0.97\(\pm 0.30\) , and \(\chi ^2=11.5/13\)  degrees of freedom (dof). The best-fit normalization in case (b) is 16.01\(\pm 0.11\) , while the best-fit slope and the best-fit \(\chi ^2\)  remain the same as in case (a). The best-fit results
are also listed in Table REF .
